Cancellation theorem for framed motives of algebraic varieties

The machinery of framed (pre)sheaves was developed by Voevodsky [V1]. Based on the theory, framed motives of algebraic varieties are introduced and studied in [GP1]. An analog of Voevodsky's Cancellation Theorem [V1] is proved in this paper for framed motives stating that a natural map of framed $S^1$-spectra $$M_{fr}(X)(n)\to\underline{\textrm{Hom}}(\mathbb G,M_{fr}(X)(n+1)),\quad n\geq 0,$$ is a schemewise stable equivalence, where $M_{fr}(X)(n)$ is the $n$th twisted framed motive of $X$. This result is also necessary for the proof of the main theorem of [GP1] computing fibrant resolutions of suspension $\mathbb P^1$-spectra $\Sigma^\infty_{\mathbb P^1}X_+$ with $X$ a smooth algebraic variety. The Cancellation Theorem for framed motives is reduced to the Cancellation Theorem for linear framed motives stating that the natural map of complexes of abelian groups $$\mathbb ZF(\Delta^\bullet \times X,Y) \to \mathbb ZF((\Delta^\bullet \times X)\wedge (\mathbb G_m,1),Y\wedge (\mathbb G_m,1)),\quad X,Y\in Sm/k,$$ is a quasi-isomorphism, where $\mathbb ZF(X,Y)$ is the group of stable linear framed correspondences in the sense of [GP1].